Int. J. Mol. Sci. 2010, 11, 4267-4284; doi:10.3390/ijms11114267 OPEN ACCESS International Journal of Molecular Sciences ISSN 1422-0067 www.mdpi.com/journal/ijms Article A Molecular–Structure Hypothesis Jan C. A. Boeyens Unit for Advanced Study, University of Pretoria, Lynnwood Road, Pretoria, South Africa; E-Mail: [email protected]; Tel.: +27-12-420-4528; Fax: +27-12-362- 5288 Received: 28 September 2010; in revised form: 25 October 2010 / Accepted: 25 October 2010 / Published: 1 November 2010 Abstract: The self-similar symmetry that occurs between atomic nuclei, biological growth structures, the solar system, globular clusters and spiral galaxies suggests that a similar pattern should characterize atomic and molecular structures. This possibility is explored in terms of the current molecular structure-hypothesis and its extension into four-dimensional space-time. It is concluded that a quantum molecule only has structure in four dimensions and that classical (Newtonian) structure, which occurs in three dimensions, cannot be simulated by quantum-chemical computation. Keywords: molecular shape; self-similarity; golden mean; logarithmic spiral; hypercomplex number; quaternion; general covalence; fibonacci phyllotaxis; solar system; atomic structure; quantum potential; space-time topology 1. Introduction Early suggestions that molecules have three-dimensional structures were greeted with derision, and recent efforts to simulate such structures quantum-mechanically have all failed. Apart from experimental success to map atomic networks in the solid state, the definition of invariant shapes, which are intrinsic to molecules, remains shaky. The inherent symmetry of molecular spectra has never been linked with theoretical certainty to rigid structure and the molecular-structure hypothesis remains essentially unsubstantiated, although millions of undisputed molecular connectivities are based directly on these unproved notions of molecular structure and symmetry. Int. J. Mol. Sci. 2010, 11 4268 The contentious issue is whether any form of molecular structure can be defined quantum-mechanically at all. The essential problem was put into eloquent perspective many years ago by Claverie [1], stating that: 00[a]lthough a molecule, considered as a set of [...] nuclei and electrons, [...] belongs to the domain of microphysics, it is well-known that [...] the behaviour of its nuclei is treated either through quantum or through classical methods according to the subject of investigation: [...] quantum treatment is used for dealing with vibrational and rotational spectra, while classical reasoning is most frequently used for dealing with conformational analysis [and] reaction mechanisms [...]. [I]t will not be very easy to rebuild some connection [...] from quantum theory toward classical theory [...] and this is a somewhat ridiculous situation since [...] classical models work perfectly for most macroscopic phenomena [...]. [S]ome kind of quantum–classical connection is strongly needed00. The author reaches the conclusion that 00 ::: quantal structure manifests its[e]lf only when some measurement takes place, while, according to the notion of classical structure, the molecule could be considered a set of point-like nuclei endowed with a well-defined position at any point, independently of any measurement00. To my mind this is a weak, and even self-contradictory, distinction between a molecule and itself. The author concludes that 00 ::: accounting for the classical behaviour [...] is a genuinely non-trivial and essentially unsolved problem which deserves specific and adequate investigation00. As recommended by a referee I insert the following brief historical account of how the debate on molecular structure has developed [2]: 00Since the late 1970s theoretical chemists, who worked hard on the development of quantum chemical models for chemical purposes, also began to question the naive reductionist, albeit common, view among western philosophers of science, according to which chemical concepts and laws could simply be derived from quantum mechanical principles. Guy Woolley, in a seminal paper [3], argued that the concept of chemical structure cannot be deduced from quantum mechanics. Hans Primas [4] devoted a whole book to the issue of reductionism, arguing that quantum mechanical holism does not allow the derivation of statements about chemical objects without further assumptions. Giuseppe Del Re and Christoph Liegener considered chemical phenomena to lie on a higher level of complexity that emerges from but does not reduce to the quantum mechanical level [5,6].00 The questionable physical grounds for the structure hypothesis are now re-examined and the conjecture that, like other objects in the universe, molecules acquire their shape in harmony with four-dimensional space-time curvature, is also explored. Int. J. Mol. Sci. 2010, 11 4269 2. Results and Discussion 2.1. Cosmic Self-Similarity The widespread perception that all structures in the cosmos share a common design principle was given some theoretical content by Oldershaw’s statement of cosmic self-similarity [7]. The idea of self-similarity is closely related to fractal geometry and the golden ratio as expressed in the properties of the golden logarithmic spiral. Oldershaw distinguishes between three hierarchical structures at the atomic or subatomic, stellar or substellar and galactic scales. The basic premise is that the fractal nature of the universe leads to self-similarity, or invariance with respect to scale transformation, in which small parts of a structure have geometrical properties that resemble the whole structure or large parts thereof. The most familiar example of self-similarity is the Mandelbrot set and the logarithmic spiral is the only smooth curve that is self-similar at all scales. Figure 1. The golden logarithmic spiral. tan α = τ. α α Self-similarity between objects on different scales implies the existence of a mathematical operation which is equally effective on different cosmic scales and which operates on symmetry elements in any number of dimensions from one to four. A quaternion is such an object, whose operation is best understood in terms of rotation in the complex plane, described by the complex number (a; b) = a + ib ≡ r(cos θ + i sin θ) = reiθ ! ! a −b cos θ − sin θ which is isomorphic with the matrix , e.g., , under addition and b a sin θ cos θ ! 0 −1 multiplication. For example, the complex number i is represented by the matrix that 1 0 corresponds to a counter-clockwise rotation of π=2 about the origin. The locus of a continuously rotating point is a circle. By also allowing continuous dilatation of the radius, the locus becomes an equiangular, or logarithmic spiral, best known in the classical form r = Aeθ cot ' (1) where A and ' are constants. It is common practice to set A = 1 to obtain a unit spiral. For ' = 72:83◦ the spiral r = Aeθτ=2 is virtually identical to a golden spiral, which develops on removing a square Int. J. Mol. Sci. 2010, 11 4270 (gnomon) from a golden rectangle and continuing the process indefinitely on the newly created golden rectangles of diminishing size, as shown in Figure 1. The inscribed golden spiral converges to the intersection of diagonals. More generally, in terms of the complex number a + ib: r = Ae(a=b)θ ; b 6= 0 It is instructive to note that rotation in the complex plane has no counterpart in three dimensions, as first discovered by William Hamilton. The direction of a rotation axis in three dimensions is perpendicular to the complex plane and provides no extra information about the rotation. An equivalent conclusion is that three-dimensional complex numbers are undefined. The simplest hypercomplex number, defined in four dimensions, is known as a quaternion: q = a + ib + jc + kd p where i, j, k are generalizations of −1, such that i2 = j2 = k2 ij = k ; jk = 1 ; ki = j ; ji = −k ; kj = −1 ik = −j and q2 = a2 + b2 + c2 + d2. A four-dimensional rotation is now defined by the quaternion Qeθ(iα+jβ+kγ) = Qfcos θ + sin θ(iα + jβ + kγ)g Whereas a complex number is defined by a vector and a phase, a quaternion is specified by a tensor and a versor. The demonstration that both Lorentz transformation and quantum spin are the direct result of quaternion rotation implies that all relativistic and quantum structures must have the same symmetry. This is the basis of cosmic self-similarity. The observation that the golden mean features in many self-similarities is interpreted to show that τ represents a fundamental characteristic of space-time curvature. The existence of antimatter and the implied CPT symmetry of space-time favours closed metric-free projective geometry with involution; the only topology that automatically generates the gauge invariance that links quantum mechanics to the electromagnetic field [8]. This topology is consistent with constant space-time curvature, locally distorted by large gravitating masses. It seems reasonable to assume that the logarithmic spiral (1) follows the general curvature in two-dimensional projection, characteristic of stable structures and growth patterns in tangent Euclidean space. In four-dimensional space-time the curvature is more appropriately described by a formula such as p ρ(xµ) = Ae(a= b2+c2+d2)θ which describes spherical rotation in quaternion notation. For our purpose all structures which can be related unequivocally in terms of the golden ratio and/or a golden spiral are assumed universally self-similar. This includes the packing of nucleons, nuclide periodicity, general covalency,